CRPE Engineer Registration cheat sheet

Reliability R(t) = probability a unit performs its required function under stated conditions for a stated time interval; it is a probability, so 0 ≤ R(t) ≤ 1 and R(0) = 1.

Failure rate λ(t) = f(t)/R(t); for the exponential model λ is constant and R(t) = e^(−λt), MTBF = 1/λ.

Bathtub curve phases: infant mortality (decreasing λ, screened by burn-in), useful life (constant λ), wear-out (increasing λ, addressed by preventive replacement).

Maintainability M(t) = probability a failed item is restored to serviceable condition within time t; repair-time density is often lognormal, not exponential.

Availability A = MTBF/(MTBF+MTTR) for a single repairable unit; inherent availability excludes logistics delay, operational availability includes it.

Series system: R_s = Π R_i and λ_s = Σ λ_i (constant-rate case); adding components always degrades series reliability.

Parallel (active redundancy) with n identical independent units: R = 1 − (1 − R_unit)^n.

MTTF applies to non-repairable items (mean of the failure distribution); MTBF applies to repairable items and counts repair cycles.

Weibull reliability R(t) = exp[−(t/η)^β]; β<1 decreasing failure rate, β=1 exponential, β>1 wear-out, β≈3.6 approximates normal.

Weibull CDF linearizes as ln[−ln R(t)] = β ln t − β ln η, so Weibull probability plots use log-log versus log axes and the slope equals β.

Characteristic life η is the time at which R = e^(−1) ≈ 0.368, independent of β; B10 life = η[−ln(0.9)]^(1/β) ≈ 0.10536^(1/β)·η.

Exponential: memoryless, constant hazard, F(t)=1−e^(−λt); mean = standard deviation = 1/λ.

Lognormal repair times: parameters μ (log-mean) and σ (log-shape); median = e^μ, mean = e^(μ+σ²/2), so mean > median.

Binomial: P(k) = C(n,k)p^k(1−p)^(n−k); used for success-run and attribute demonstration tests with fixed trials.

Poisson: P(k failures in t) = (λt)^k e^(−λt)/k!; models counts of failures in time when interarrival times are exponential.

Confidence interval for constant failure rate from time-terminated test: 2T/χ²(1−α/2, 2r+2) ≤ 1/λ ≤ 2T/χ²(1−α/2, 2r) with T total test time and r failures.

Derating: operate parts below rated stress (power, voltage, temperature, current) to lower failure rate; derating curves come from part stress versus failure-rate data.

Parts-count prediction: λ_system = Σ N_i λ_i π_Q,i (π_Q = quality factor), used early when detailed stress data are unavailable.

Stress-strength interference: failure occurs when stress exceeds strength; increase margin or reduce variability of either distribution to lower failure probability.

Redundancy types: active (parallel, all powered), standby (cold/warm, switched), and voting (k-out-of-n); k-out-of-n with identical units R = Σ C(n,i) R^i (1−R)^(n−i) for i = k to n.

Common-cause failure defeats redundancy; model with β-factor: system failure rate ≈ β·λ + (independent term); physical separation and diversity reduce β.

Thermal design: every 10 °C rise roughly halves many electronic failure rates (Arrhenius acceleration with activation energy Ea, AF = exp[(Ea/k)(1/T_use − 1/T_stress)]).

FMEA and derating reviews are design-phase tools; robustness (Taguchi) reduces sensitivity to noise factors rather than removing them.

Design reviews should verify: no single-point failures for critical functions, adequate margins, and part stress ratios within program derating guidelines.

Success-run (zero-failure) demonstration: n = ln(1−C)/ln(R) units tested to time t with no failures, for confidence C and reliability R.

Exponential test time for a fixed number of allowed failures uses the chi-square relationship: T = χ²(1−C, 2r+2)·θ/2 for zero-failure or r-failure plans.

Sequential (probability-ratio) tests use producer's risk α, consumer's risk β, and discrimination ratio to set accept/reject lines; they typically reduce expected test time.

Accelerated life testing uses an acceleration model (Arrhenius for temperature, inverse power law for voltage, Coffin-Manson for thermal cycling) to extrapolate use-life.

Environmental stress screening (ESS) is a manufacturing screen (thermal cycling, vibration, burn-in) that precipitates latent defects; it is not a reliability demonstration.

Highly accelerated life testing (HALT) finds operating and destruct limits; highly accelerated stress screening (HASS) applies the resulting limits in production.

Test plans must state the statistical model, the discrimination ratio, the risks, and the failure definition before testing starts; post-hoc changes invalidate the demonstration.

Degradation testing measures a parameter drift over time and predicts failure at a threshold, useful when failures are rare within test duration.

FMECA extends FMEA by adding criticality; item-level criticality Cm = β·α·λ·t, where β is conditional probability the failure causes the effect, α the mode ratio, λ the item failure rate, t the operating time.

Mode criticality number Cr = Σ over effects of (β·α·λ·t); item criticality sums mode criticalities.

RPN = Severity × Occurrence × Detection (1–10 scales); higher RPN ranks higher priority, but severity should also act as a gate regardless of RPN.

Severity, occurrence, and detection scales must be defined in the analysis ground rules; inconsistent scales make RPN comparisons meaningless.

FMECA types: system/functional (top-down, early design) and hardware/part (bottom-up, detailed design); design FMECA addresses design causes, process FMECA addresses manufacturing causes.

Action thresholds: any mode with severity 9–10 and no detection control requires design action even if RPN is moderate.

Failure mode ratio α comes from field or test data; when unknown, engineering judgment must be documented as an assumption.

FMECA is a living document: update after design changes, test findings, and field failures; it feeds the critical items list and test planning.

Fault tree gates: AND gate output occurs only if all inputs occur (corresponds to parallel redundancy in the success path); OR gate occurs if any input occurs (corresponds to series).

Minimal cut sets are the smallest combinations of basic events that cause the top event; single-element cut sets are single points of failure and must be resolved for critical functions.

Rare-event approximation: P(top) ≈ Σ P(cut set) when basic event probabilities are small and cut sets are not strongly overlapping.

Boolean reduction: RBD series = AND in the failure tree; RBD parallel = OR in the failure tree. Convert one to the other before computing.

k-out-of-n in an RBD: success requires at least k of n paths; reliability uses the binomial sum with identical units.

Cut-set importance measures (Birnbaum, Fussell-Vesely) rank basic events by contribution to top-event probability and guide where to spend reliability effort.

Fault tree construction rules: define the top event precisely (what, where, when), use immediate causes, and avoid bypassing intermediate events.

RBD assumes independent failures unless common-cause is explicitly modeled; shared power, cooling, or software invalidates the independence assumption.

MTTR is the mean of the repair-time distribution; for lognormal repair times use the lognormal mean, not the median, when computing availability.

Inherent availability A_i = MTBF/(MTBF+MTTR); achieved availability adds preventive and corrective maintenance time; operational availability adds logistics and administrative delay.

Maintainability demonstration: measure repair times against a stated percentile (e.g., 90% of repairs within 30 minutes) using a lognormal or chi-square based plan.

Maintenance concepts: corrective (run-to-failure), preventive (time-based), predictive (condition-based), and detective (inspection); choose by failure consequence and P-F interval.

P-F interval is the time between potential-failure detection and functional failure; inspection frequency should be a fraction of the P-F interval.

Repair-rate μ = 1/MTTR; two-state availability for a single unit with constant λ and μ is A = μ/(λ+μ).

Maintainability design features that reduce MTTR: accessibility, modularity (LRU replacement), built-in test, standardization, and unambiguous maintenance procedures.

Spare parts and test equipment availability affect operational availability but not inherent availability; state which availability is being specified.

Duane model: cumulative MTBF = A·T^α; instantaneous MTBF = A·T^α/(1−α); plot cumulative MTBF versus cumulative test time on log-log axes.

AMSAA/Crow model fits cumulative failures N(t) = λ t^β; growth occurs when β<1, and the instantaneous failure intensity is λβ t^(β−1).

Growth planning uses the growth rate α (or β) to project test time needed to reach a target MTBF from the current demonstrated value.

Life data analysis steps: choose distribution (Weibull, lognormal, exponential), estimate parameters (MLE or rank regression), check fit, then compute metrics such as B10 life.

Censored data: right-censored (suspended) units contribute time without failure and must be included; ignoring them biases MTBF downward.

Median ranks for plotting: F = (i−0.3)/(n+0.4) (Benard) or (i−0.5)/n; choice affects plotted positions, not the underlying data.

Failure modes must be separated before fitting; mixed modes (e.g., infant mortality plus wear-out) produce a poor single-Weibull fit.

Goodness-of-fit: use probability-plot correlation coefficient, Anderson-Darling, or likelihood ratio to compare candidate distributions before extrapolating.

Risk = probability of occurrence × severity of consequence; risk matrices map likelihood and severity to acceptance categories (acceptable, tolerable, unacceptable).

ALARP principle: risk must be reduced as low as reasonably practicable; demonstrate that further reduction cost is grossly disproportionate to benefit.

Human error probability (HEP) from THERP-style tables; human reliability is influenced by stress, training, procedure quality, and time available.

Human factors design rules: error-proofing (poka-yoke), forcing functions, clear feedback, and separation of controls reduce slips and mistakes.

Safety integrity levels (SIL) in IEC 61508 map to target average probability of failure on demand (PFDavg) ranges; SIL 4 is the most stringent.

Common-cause and systematic failures dominate safety instrumented systems; redundancy alone does not achieve high SIL without diagnostics and proof-test coverage.

Fault tolerance versus fail-safe: fail-safe designs direct failure to a safe state; fault-tolerant designs continue operation, which is not always safer.

Risk acceptance must be documented with rationale; hazard analysis (PHA, HAZOP, FTA) should trace each hazard to a mitigation and verification.

Reliability program plan defines tasks, schedule, responsibilities, and metrics; it must be tailored to the program's criticality, cost, and schedule constraints.

Reliability metrics: MTBF, failure rate, B10 life, availability, and mission reliability; each must be tied to a verification method and a data source.

FRACAS (failure reporting, analysis, and corrective action system) closes the loop from failure to root cause to verified corrective action.

Reliability growth testing, ESS, and FMECA should be scheduled in the design phase; demonstration testing occurs after design maturity.

Supplier management: flow down reliability requirements, require failure-rate data and FMECA from suppliers, and audit their test evidence.

Design reviews at preliminary and critical design gates should include reliability, maintainability, safety, and human factors as formal agenda items.

Metrics must be defined before data collection; changing the failure definition or the time base mid-program invalidates trend analysis.

Program risk register should include reliability shortfalls as risks with owners, mitigation plans, and trigger thresholds.